Supplier Selection JurInt 3

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Supplier selection using fuzzy AHP and fuzzy multi-objective linear

programming for developing low carbon supply chain

Krishnendu Shaw a,, Ravi Shankar a, Surendra S. Yadav a, Lakshman S. Thakur b

a Department of Management Studies, Viswakarma Building, Indian Institute of Technology Delhi, New Delhi 110016, Indiab Operations and Information Management Department, School of Business, University of Connecticut, 2100 Hillside Road, Storrs, CT 06269-1041, USA

a r t i c l e i n f o

Keywords:

Supplier selection

Fuzzy multi-objective linear programming

Fuzzy AHP

Carbon emission

Green house gas

Textile supply chain

a b s t r a c t

Environmental sustainability of a supply chain depends on the purchasing strategy of the supply chain

members. Most of the earlier models have focused on cost, quality, lead time, etc. issues but not given

enough importance to carbon emission for supplier evaluation. Recently, there is a growing pressure

on supply chain members for reducing the carbon emission of their supply chain. This study presents

an integrated approach for selecting the appropriate supplier in the supply chain, addressing the carbon

emission issue, using fuzzy-AHP and fuzzy multi-objective linear programming. Fuzzy AHP (FAHP) is

applied first for analyzing the weights of the multiple factors. The considered factors are cost, quality

rejection percentage, late delivery percentage, green house gas emission and demand. These weights

of the multiple factors are used in fuzzy multi-objective linear programming for supplier selection and

quota allocation. An illustration with a data set from a realistic situation is presented to demonstrate

theeffectiveness of the proposedmodel. Theproposed approach canhandle realistic situation when there

is information vagueness related to inputs.

2012 Elsevier Ltd. All rights reserved.

1. Introduction

Supplier selection plays an important role to make a supply

chain green (Rao, 2002). A positive relation between green supplier

selection and green supply chain implementation has been ob-

served in a study ofSeuring and Mller (2008). Many researchers

have addressed supplier selection issue in the green supply chain

fromthe perspectives of environmental sustainability (Bai & Sarkis,

2010; Enarsson, 1998; Handfield, Walton, Sroufe, & Melnyk, 2002;

Humphreys, Wong, & Chen, 2003a; Humphreys, McIvor, & Chan,

2003b; Hsu & Hu, 2009; Lee, Kang, Hsu, & Hung, 2009a; Noci,

1997; Rao, 2005; Walton, Handfield, & Melnyk, 1998). However,

very few studies have addressed the carbon emission and the

related issues for supplier evaluation. Recently, Lash and Welling-

ton (2007)have discussed the impacts of climate change over the

business operations. They suggested that companies have to

handle climate change risk properly for gaining the competitive

advantage. Some leading companies have already started working

to develop next generation carbon emissions management for their

supply chain to survive in the business.

An interesting survey conducted by a consulting company (Tru-

cost, 2009) showed that only 19 percent of the total green house

gas (GHG) emission in the supply chain is generated from direct

operational activities of the company and rest of the 81 percent

emission is generated from other indirect activities such as, emis-

sion from first tier supplier, electricity supplier and emission from

other supply chain members. In this scenario, supplier selection

plays an important role to minimize carbon emission in supply

chain. According to a survey reportCDP (2010), more than half of

the participants said that in the future they would cease business

with the suppliers, if they do not manage their carbon emissions.

Due to increase consciousness about climate change, companies

are imposing pressure on their suppliers to manage their GHG

emissions as one of the conditions for doing business with them.

Supplier propensity to minimize green house gas emission is

becoming one of the criteria for supplier selection (CDP, 2010).

Therefore, suppliers need to make a thorough assessment of their

current capabilities in terms of carbon emission management

and set appropriate targets for further reduction of their emissions.

WalMart in US can be taken as an example of a global supply

chain which has been trying to achieve environmental sustainabil-

ity. Its aim is to become a world leader in environmental sustain-

ability. To achieve this, it has suggested the suppliers to reduce

their energy consumption for processing of products (WalMart,

2010). Suppliers who measure and publish their own emission

are strategically more preferable than others because they help

the buyers to manage their carbon emission. However, only a little

number of supply chain members has extensive knowledge about

0957-4174/$ - see front matter 2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.eswa.2012.01.149

Corresponding author. Tel.: +91 9999841552; fax: +91 11 26862620.

E-mail addresses:[email protected],[email protected]

(K. Shaw).

Expert Systems with Applications 39 (2012) 81828192

Contents lists available at SciVerse ScienceDirect

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the low-carbon material procurement for their supply chain. This

paper deals with the low-carbon material procurement and carbon

management for supplier selection. The relevant literature on

green supplier selection is discussed below.

Noci (1997)proposed a green vendor rating framework for the

assessment of suppliers environmental performance. Green com-

petence, green image, life cycle cost, and environmental efficiency

were the important considerations for supplier evaluation.

Humphreys et al. (2003b) developed a knowledge-based system

to evaluate suppliers environmental performance. Cost, manage-

ment competencies, green image, green design, environmental

management system, and environmental competencies were con-

sidered as the evaluation factors in the model. Lu, Wu, and Kuo

(2007) proposed an analytic hierarchy process (AHP) and fuzzy

logic based model for Green supplier evaluation. Further, analytic

network process (ANP) based framework was suggested by Hsu

and Hu (2009) to construct an assessment framework of the

supplier for Taiwanese Electronics Company. Five criteria such as

procurement management, R& D management, process manage-

ment, incoming quality control, and management system were

considered in the model.Lee et al. (2009a)suggested an integrated

model to select green suppliers for high-tech industry considering

six factors. The considered factors were quality, technology capa-

bility, pollution control, environmental management, green prod-

uct, and green competencies.

Bai and Sarkis (2010) developed a green supplier evaluation

model considering economic, environmental, and social issues.

Rough set theory was used to deal with the information vagueness.

Kuo, Wang, and Tien (2010) developed a green supplier selection

model applying artificial neural network (ANN) and two multi-

attribute decision analysis (MADA) methods that consists of data

envelopment analysis (DEA) and analytic network process (ANP).

Awasthi, Chauhan, and Goyal (2010) developed a fuzzy multi-crite-

ria model for evaluating environmental performance of suppliers.

Fuzzy TOPSIS was applied in this model. Buyukozkan and Cifci

(2011) proposed fuzzy multi-criteria decision framework for sus-

tainable supplier selection considering incomplete information.Fuzzy analytic network process within the multi-person deci-

sion-making scheme under incomplete preference relations was

used in their model.

Earlier studies have limited focus on the carbon management is-

sue for supplier evaluation. Earlier studies have mostly focused on

multi-criteria decision making approaches such as AHP, fuzzy AHP,

fuzzy ANP, TOPSIS, Rough set theory etc. for supplier evaluation.

These types of the models are less robust because quantification

of order quantity to a particular supplier is not possible. To solve

this drawback a hybrid model using fuzzy AHP, fuzzy linear pro-

gramming is proposed forselection of supplier. In fewof these stud-

ies, product carbon footprint is taken as one of the criteria of

supplier selection. Product carbon footprint can be measured by

usingPublicly Available Specification (PAS) 2050 (2008) standarddeveloped by British Standard Institution. The buyer can fix certain

amount of carbon emission cap, which acts as a constraint in the

decision model. The presentarticle is organized as follows. Section 2

explores the literature related to supplier selection methodologies.

Section3discusses the fuzzy set theory. In Section 4, multi-objec-

tive mathematical model is shown. Section 5 represents case study,

related results and discussions. Section6presents the conclusions.

2. Supplier selection problem

Business environment is continuously changing due to diversifi-

cation of customer demands. This diversification of demand leads

to increase in operating cost and followed by the decrease in profit.Therefore, purchasing decisionfrom a particular supplieris a crucial

strategic decision to ensure profitability and long term survival of

the company. Most of the companies are trying to reduce their

operating costs while satisfying customer needs by increasing their

core competencies and outsourcing other functions (Lee, 2009). A

careful assessment is needed to select right supplier who can main-

tain a continuous replacement of product in proper time. Most of

the times supplier strength and weakness are varied, which leads

to complex decision making of supplier selection. Many researches

in supplier selection area used mathematical programming.

Ghodsypour and OBrien (1998) solved a supplier selection prob-

lemusinga hybrid approach involvingAHPand linear programming.

A mixed integer non-linear programming model considering multi-

ple sourcing opportunities was solved byGhodsypour and OBrien

(2001). Total costof logisticswith budgetconstraint, quality,service,

etc. were considered in their model.Karpak, Kumcu, and Kasuganti

(1999)proposed a goal programming model that minimized costs

and maximizeddelivery reliability and quality for supplierselection

and quota allocation.

Gao and Tang (2003) formulated a multi-objective linear pro-

gramming model to purchase raw materials for a large-scale steel

plant in China.Kumar, Vrat, and Shankar (2004)developed a fuzzy

goal programming approach for vendor selection considering the

effect of information uncertainty in the decision making. Similar

type of problem was solved by Amid, Ghodsypour, and OBrien

(2006). They used fuzzy multi objective linear programming to

determine the order quantity from many suppliers by considering

the criteria of lowest cost and highest quality. Hong, Park, Jang, and

Rho (2005)proposed a mathematical model for supplier selection

considering the change in suppliers supply capabilities and

customers needs over a period of time. This model optimized the

revenue and customer needs simultaneously.

There are numerous studies, which applied the dual methodol-

ogies for supplier selection.

Weber, Current, and Desai (2000)formulated a combined multi

objective programming (MOP) and the DEA based framework for

supplier selection. They applied MOP to calculate the order quan-

tity and used DEA for suppliers efficiency evaluation. Further,Cebiand Bayraktar (2003) solved a supplier order allocation problem

considering the quantitative as well as qualitative criteria. Wang,

Huang, and Dismukes (2004) applied AHP method to choose a

strategy from agile/lean supply chain. Further, they used pre-emp-

tive goal programming (PGP) to obtain the optimal order quantity

from the suppliers.

Chan and Kumar (2007) developed a fuzzy extended analytic

hierarchy process (FEAHP) model for global supplier selection. Fur-

ther,Kumar, Shankar, and Yadav (2008)solved a supplier selection

problem using AHP and fuzzy linear programming. Ku, Chang, and

Ho (2010)andLee, Kang, Hsu, and Hung (2009b) used fuzzy AHP

and fuzzy goal for supplier selection. In their model, fuzzy AHP

was applied first to calculate the weights of the criteria. The crite-

rias weights were subsequently used in fuzzy goal programmingto select the supplier. Amin, Razmi, and Zhang (2011) developed

a supplier selection model using fuzzy SWOT analysis and fuzzy

linear programming. Ycel and Gneri (2010) proposed a weighted

additive fuzzy programming approach for supplier selection. They

used TOPSIS and fuzzy linear programming in their framework.

3. Fuzzy set theory

Decision making is very difficult for vague and uncertain envi-

ronment. This vagueness and uncertainty can be handled by using

fuzzy set theory, which was proposed by Zadeh (1965). Fuzziness

and vagueness are normal characteristics of a decision making

problem. This fuzziness and vagueness can be managed by increas-ing robustness of the model (Yu, 2002). If we do not consider the

K. Shaw et al. / Expert Systems with Applications 39 (2012) 81828192 8183


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fuzziness during the decision making process then the results

evolved from the model may mislead the decision maker. Fuzzy

theory is very useful to solve such practical problems.

Many times decision makers provide an uncertain answer

rather than a precise value. Therefore, it is very difficult to quantify

this qualitative value (Lee, Kang, & Wang, 2005). In AHP, the crisp

value is taken for the pair-wise comparison but this method is not

appropriate for real life decision making problem where fuzziness

is present. To solve this problem, a degree of uncertainty is to be

considered in the decision model (Lee, 2009; Yu, 2002). Incorpora-

tion of the fuzzy theory in AHP is more appropriate and more effec-

tive than conventional AHP. In fuzzy AHP, the concept of fuzzy set

theory is used, and calculation is done as per normal AHP method

for selecting the alternatives (Bozbura, Beskese, & Kahraman,

2007). There are many areas where fuzzy AHP has been applied

for decision making; and many researchers have developed differ-

ent methodologies for calculating the fuzziness (Boender, DeGraan,

& Lootsma, 1989; Buckley, 1985; Chen, 1996; Chang, 1996; Csutora

& Buckley, 2001; Laarhoeven & Pedrycz, 1983; Lee et al., 2005; Lee,

2009). There are many different methods available for ranking of

the fuzzy number but every method has its own advantage and

disadvantage (Klir & Yan, 1995).

Lee and Li (1988) proposed intuition ranking method that calcu-

lates the ranks of triangular fuzzy numbers by drawing their mem-

bership function curve. Adamo (1980), Yagar (1978) proposed

/-cut method and centroid method respectively to rank the fuzzy

numbers. The extent analysis method proposed by Chang (1996)is

applied here because the computation is much easier than other

fuzzy AHP processes and it takes little time to calculate. Another

advantage of this method is that it can overcome the deficiencies

of conventional AHP process. This fuzzy AHP not only handles

the uncertainty imposed by the decision maker during decision

making process, but it also provides the robustness and flexibility

during the decision making (Chan & Kumar, 2007). Triangular fuz-

zy number is used to calculate the priority of different decision var-

iable by pair-wise comparison, and the extent analysis is used to

calculate the synthetic value from pair-wise comparison. A briefintroduction of the fuzzy set theory is given below. Triangular

fuzzy number is used extensively for most of the fuzzy applica-

tions. A triangular fuzzy number M^

is shown in Fig. 1. A fuzzy num-

ber can be represented by (a, b, c) and the membership function can

be defined as follows(1)(Cheng, 1999; Lee et al., 2005; Lee, 2009).

lM^ x

xaba

a 6 x 6 bcxcb

b 6 x 6 c

0 Otherwise

8>: 1with 1 < a6 b6 c61.

The strongest grade of membership is the parameter b that is,

fM(b) = 1 whilea andcare the lower and upper bounds. Two trian-

gular fuzzy number M1 m1 ;m1;m

1

and M2 m2 ;m2;m

2

shown in

Fig. 2is compared byLee et al. (2009a).

when m1 P m2; m1 P m2; m

1 P m

2 2

The degree of the possibility is defined as (3):

VM1 P M2 1 3

Otherwise, the ordinate of the highest intersection point is calcu-

lated as (Chang, 1996; Lee, 2009; Zhu, Jing, & Chang, 1999)

VM2 P M1 hgtM1 \ M1 ld m1 m

2

m2m2

m1m

1

4

The value of the fuzzy synthetic extent can be calculated as follows

(5)(11)(Chang, 1996; Lee, 2009; Zhu et al., 1999).

Fi Xm

j1

Mjgi

Xn

i1 Xm

j1

Mjgi" #

1

; i 1;2;. . .;n 5

Xmj1

Mjgi Xmj1

mij;Xmj1

mij;Xmj1

mij

!; j 1; 2;. . .;m 6

Xni1

Xmj1

Mjgi

" #1

1Pni1

Pmj1M

ij

; 1Pni1

Pmj1Mij

;1Pn

i1

Xmj1

Mij

! 7

A convex fuzzy number can be defined as,

VFP F1; F2;. . .;Fk minVFP Fi; i 1; 2;. . .;k 8

dFi minVFi P Fk W0i k 1;2;. . .;n and k i 9

Based on the above procedure, the weights, W0i of factors are

W

0

W0

1; W

0

2;. . .;W

0

n T 10After normalization, the priority weights are as follows

W W1; W2;. . .;WnT

11

4. Supplier selection model

The following sets of assumptions, index set, decision variable

and parameters are considered for formulating the multi-objective

supplier selection model.

(i) Only one type of product is purchased from one supplier.

(ii) This model does not consider quantity discounts.

(iii) No shortage of the item is allowed for any of the suppliers.(iv) It is assumed that lead time is constant.Fig. 1. Triangular fuzzy number.

Fig. 2. Two triangular fuzzy numbers M1 and M2 (Lee, 2009).

8184 K. Shaw et al. / Expert Systems with Applications 39 (2012) 81828192


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Index

i index for suppliers, for all i = 1,2,. . . , n.

j index for objectives, for all j = 1,2,. . . ,J.

k index for constraints, for all k = 1,2,. . . , K.

Decision variable

xi order quantity given to the supplier i.

Parameters

D aggregate demand of the item over a fixed planning period.n number of suppliers competing for selection.

Pi price of unit item of the ordered quantity xi to the supplier i.

Qi percentage of the rejected units delivered by the supplier i.

Li percentage of the late delivered units by the supplier i.

Gi green house gas emission (GHGE) for product supplied by

supplier i.

Ui upper limit of the quantity available for the supplier i.

Bi budget constraint allocated to supplier i.

Ccap total carbon emission cap for sourcing of material.

Model

A typical linear model for supplier selection problem can be for-

mulated as follows (Amid et al., 2006; Kumar, Vrat, & Shankar,

2006):

Minimise Z1Xni1

Pixi 12

Minimise Z1Xni1

Qixi 13

Minimise Z3Xni1

Lixi 14

Minimise Z4Xni1

Gixi 15

Subject to;

Xn

i1

xi D 16

xi 6 Ui 17Xni1

Gixi 6 Ccap 18

Pixi 6 Bi 19

xi P 0 and integer 20

Objective function(12)minimizes the total cost of ordering.

Objective function (13) minimizes the rejection due to the qual-

ity problem.

Objective function(14)minimizes the late delivered items of

the suppliers.

Objective function (15) minimizes the total green house gas

emissions for procurement.Constraint(16)shows the total aggregate demand of the item.

Constraint (17) ensures the maximum available capacities of

the suppliers.

Constraint (18) puts restrictions on carbon footprint for

sourcing.

Constraint (19) puts restrictions on the budget amount allo-

cated to the suppliers for supplying the items.

Constraint(20) ensures all the variables greater than zero and

integer.

In real life problem of supplier selection, there are many factors,

which are not known properly, create vagueness in the decision

environment. This vagueness cannot be interpreted by the deter-

ministic problem. Therefore, the deterministic models are not suit-able for real life problems (Kumar et al., 2006). Fuzzy technique is

applied here to deal with the problem. In this method, it is desired

to maximize the overall aspiration level rather than strictly satisfy-

ing the constraints (Kumar et al., 2006).

Zimmermann (1978) developed a multi objective fuzzy linear

programming, which can handle linguistics issues properly in

decision making. Fuzzy decision is classified into two categories,

symmetric and asymmetric fuzzy decision-making. In symmetrical

fuzzy decision same weights are considered for objectives and con-

straints, but in case of asymmetric fuzzy decision-making, the

weights are different for objectives and constraints (Sakawa,

1993; Zimmermann, 1978). Multi-objective programming consid-

ering the fuzzy goals and fuzzy constraints can be transformed into

crisp linear programming formulation (Zimmermann, 1978). We

have adopted the weighted additive model proposed by Tiwari,

Dharmahr, and Rao (1987) for computing the supplier selection

problem because in the real situation all the objective functions

and constraints have different weights. The weights have been cal-

culated by using fuzzy AHP extent method proposed by Chang

(1996).

4.1. Fuzzy linear programming

Fuzzy linear programming was proposed by Zimmermann

(1978). Fuzzy linear programming consists of fuzzy goals, and fuz-

zy constraints can be reformulated in such a way that it can be

solved like a normal linear programming problem.

Conventional LP problem proposed by Zimmermann (1978)is

given below (21)(23).

Minimise Z Cx 21

Subject to

Ax 6 b 22

x P 0 23

After fuzzification the equation can be represented like this(24)

(26),

Cx -Z 24

eAx - b 25x P 0 26

The symbol- in the constraint set denotes essentially smaller than

or equal to and allows one reach some aspiration level where CandA represent the fuzzy values.

4.2. Membership function

Fuzzy set was proposed by Bellman and Zadeh (1970). The fuzzy

setA in Xis defined as(27):

A fx;lAx=x2 Xg 27

wherelA(x):x? [0,1] is called the membership function ofA andlA(x) is the degree of membership to which xbelongs toA. The fuz-zy set A is thus uniquely determined by its membership function

lA(x) and the range of membership function is a subset of thenon-negative real numbers whose value is finite and usually finds

a place in the interval [0,1].

A linear membership function has been considered in this mod-

el for all fuzzy parameters. A linear membership function has char-

acteristics of continuously increasing or decreasing value over the

range of the parameter. It is defined by the lower and upper values

of the acceptability for that parameter.

A fuzzy objective Z2Xis a fuzzy subset ofXcharacterized by its

membership function lA(x): x?

[0,1]. The linear membershipfunction for the fuzzy objectives is given as:

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


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lZjx

1 if Zjx 6 Zminj

Zmaxj

Zjx Zmax

j Zmin

j if Zminj 6 Zjx 6 Z

maxj

0 if Zjx PZmaxj

8>>>>>: where j 1; 2;. . .;J:28

In(28)Zmin

j isminjZj(x) andZ

maxj ismaxjZj(x

) andx is the optimum

solution.A fuzzy constraint C2Xis a fuzzy subset ofXcharacterized by

its membership functionlC(x):x? [0, 1]. The linear membershipfunction for the fuzzy constraints is given by(29):

lCk x

1 if gkx 6 bk;

1 fgkx bkg=dk if bk6 gkx 6 bkdk;

0 if bkdk 6 gkx

8>: 29for all fuzzy parametersk = 1,2,. . . , K. The interpretation ofdkis the

tolerance interval.

4.3. Solution of the formulation

A fuzzy solution is the intersection of all the fuzzy sets repre-

senting either fuzzy objectives or fuzzy constraints (Bellman & Za-

deh, 1970). The membership function of the fuzzy solution is

represented by(30).

lSx lZx \ lCx minlZx;lCx 30

In the Eq. (30) lZ(x), lC(x) and lS(x) represent the membershipfunctions of objectives, constraints and solutions, respectively.

The fuzzy solution of the supplier selection model for theJfuzzy

multiple objectives andKconstraints may be represented as(31),

lSx \Jj1

lZx

!\

\Kk1

lCx

!

min minj1;2;...;J

lZj x; mink1;2;...;KlCk x 31

Highest degree of the membership value is the optimum solution of

the supplier selection problem(32).

lsx max

x2SlSx max

x2xmin min

j1;2;...;JlZj x; mink1;2;...;K

lCk x

32

4.4. Crisp formulation of the supplier selection model

A fuzzy programming model consists ofJobjectives andKcon-

straints are transformed into the following crisp formulation.

Crisp formulation can be represented by (33) and (38)(Kumar

et al., 2006),

Maximise k 33

Subjected to;

k Zmaxj Zminj

Zjx 6 Z

maxj for allj; j 1; 2;. . .;J 34

kdx gkx 6 bkdk for allk; k 1;2;. . .;K 35

Ax 6 b for all the deterministic constant; 36

x P 0 and integer 37

0 6 k 6 1 38

According to theZimmermann (1978) the optimum lower bound

Zminj

and upper boundZ

maxj can be calculated by solving the same

objective function two times like minimization and maximization

respectively.

The lower bound of the optimal values Zminj

is obtained by

solving the supplier selection problem as a linear programmingproblem(39)(42).

Minimise Zjx for allj; j 1;2;. . .;J 39

Subjected to

gkx 6 bkdk for allk; k 1; 2;. . .;K 40

Ax 6 b for all the deterministic constant 41

x P 0 and integer 42

The upper bound of the optimal values Zmax

j is obtained by solving

a similar supplier selection problem as a linear programming prob-

lem(43)(46).

Maximise Zjx for allj; j 1;2;. . .;J 43

Subjected to

gkx 6 bkdk for allk; k 1; 2;. . .;K 44

Ax 6 b for all the deterministic constant 45

x P 0 and integer: 46

According to Zimmermann (1978), the weight of the objective func-

tions and constraints are same in the crisp formulation of the sup-

plier selection problem. However, for a real life supplier selection

problem, all objective functions cannot be given same weights. By

using same weights, the value of important objective function is

decreased. As a result, an optimal solution for supplier selection

may not be obtained for that case. To avoid this problem, we are

adopting the weighted additive model. The weighted additive mod-

el is widely used in multi objective optimization problems. Linear

weighted utility function is obtained by multiplying each member-

ship function of fuzzy goals by their corresponding weights and

then adding the results together.

The weighted additive model proposed by Tiwari et al. (1987) is

lDx XJj1

wjlzj x XKk1

bklgk x 47

XJ

j1

wj XK

k1

bk 1; wj; bk P 0 48

In(47) and (48),wj andbk are the weights coefficients that present

the relative importance among the fuzzy goals and fuzzy

constraints.

The following crisp single objective programming(49)(55)is

equivalent to the above fuzzy model.

MaximiseXJj1

wjkjXKk1

bkck 49

Subject to;

kj 6 lzj x; j 1;2;. . .;J 50

ck 6 lgk x; k 1;2;. . .;K 51

gpx 6 bp; p 1;. . .;M 52kj; ck 2 0; 1; j 1; 2;. . .;Jand k 1;2;. . .;K 53XJj1

wjXKk1

bk 1; wj; bk P 0 54

xi P 0; i 1; 2;. . .;n 55

4.5. Application of fuzzy linear programming for supplier selection

The fuzzy linear programming for supplier selection is proposed

below. In this mathematical model, we are considering cost,

rejection percentage, late delivery percentages, green house gas

emission per product and demand are fuzzy information. After

fuzzification, we can represent the equations as follows(56)(64)(Kumar et al., 2006):



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Xni1

Pixi -fZ1 56Xni1

Qixi -fZ2 57

Xn

i1

Lixi-

fZ3 58

Xni1

Gixi- fZ4 59Xni1

xi ffi D 60

xi 6 Ui 61Xni1

Gixi 6 Ccap 62

Pixi 6 Bi 63

xi P 0 and integer 64

4.5.1. Computational procedure

In this study, a combined approach of fuzzy-AHP and fuzzymulti-objective linear programming is used to solve the problem.

Fuzzy-AHP is used to determine the relative weights of supplier

selection criteria. These weights are multiplied with each member-

ship function of fuzzy linear programming to formulate the crisp

equation. By using fuzzy-AHP, we can calculate the relative

weights of each membership function of fuzzy goals in the

different strategic environment (Ku et al., 2010).

The solution steps to solve this model are given below.

Step 1: Identification of supplier selection criteria is done first.

Step 2: Questionnaire is developed for pair wise comparison of

factors. Experts in the fields of supply chain and operations

management were asked to fill the nine-point-scale ques-

tionnaire. The consistency property of each experts com-parison results has to be checked first. If there is any

inconsistency, the questionnaire is to be filled again and

the whole process is to be repeated until the consistency

requirement is met.

Step 3: Fuzzy importance weight of the criteria is calculated using

the response of the experts. A triangular fuzzy number D is

obtained by combining the experts opinions (Lee, 2009).

D h;h;h

where

h

Ys

t1

lt

!1=s; 8t 1; 2;. . .; s:

hYst1

mt

!1=s; 8t 1; 2;. . .; s:

h

Yst1

ut

!1=s; 8t 1; 2;. . .; s:

and (lt, mt, ut) is the importance weights from expert t.

Step 4: Crisp relative importance weight (priority vectors) for fac-

tors is calculated using the extent analysis method (EAM)

proposed by Chang (1996). By using, Eqs. (2)(11), the

weights of the factors are calculated.

Step 5: Supplier selection objective functions are formulated.

These objective functions are cost minimization, rejection

minimization, late delivery minimizations and green-

house gas emission minimization.

Step 6: The first objective is selected and solved. After solving the

first objective, we obtained the lower bound optimal value

of first objective.

Step 7: The process is repeated for the remaining objectives one

by one. The lower bound and upper bound for each of

the objectives are calculated using the same set of

constraints.

Step 8: The crisp formulation is done using the weighted additive

model proposed by Tiwari et al. (1987). The weights of the

factors which are calculated earlier by EAM are used to

formulate the crisp formulation.

Step 9: The crisp formulation of the fuzzy optimization problem is

solved and result is obtained.

5. A case illustration

The effectiveness of the model is discussed through a case con-

ducted for an Indian based garment manufacturing company

(ABC). The company is fully export oriented and fulfills demand

of American and European customers. It produces a variety of gar-

ments such as Jeans, T-Shirt, formal shirting, suiting and ladies

garments, etc. The company procures finished fabric from different

suppliers and transformed it into garments in Delhi based plant in

India. Subsequently finished and packed garments are exported to

American and European markets.

The company operates in pull based system and procurement

of raw material starts after placement of order by buyers. Most

of the foreign buyers prefer to buy green and environmental sus-

tainable products. To fulfill the demand of the customers, the

management of ABC Company has decided to incorporate envi-

ronmental criteria into their suppliers evaluation process. Man-

agement has wanted to improve the environmental efficiency

and cost effectiveness of the purchasing process. The manage-

ment has realized that a loyal supplier manufacturer relationship

is needed to minimize the carbon footprint of sourcing. The rela-

tionship should be such that suppliers would share the informa-

tion with manufacturer regarding the carbon footprint of theirmanufactured product. Management has formed a special com-

mittee that consists of managers from different departments

such as purchasing, production, marketing, quality control, re-

search and development. The aim of the committee is to find

out the best supplier.

The committee has decided to take four criteria such as cost,

quality rejection, percentage of late delivered item and green

house gas emission per product for supplier selection. After decid-

ing the factor the committee has chosen four potential suppliers

for sourcing of the material. After deciding the selection criteria,

a brain-storming session was conducted in the presence of pur-

chasing and operations managers to prioritize these criteria by

using the FAHP method. The membership functions of triangular

fuzzy numbers are given in Table 1 (Lee, 2009). The fuzzy pair wisecomparisons among the criteria are shown inTable 2.

Table 1

Characteristic function of the fuzzy numbers.

Fuzzy number Characteristic (membership) function

~1 (1,1,2)

~x (x 1,x,x+ 1) forx = 2, 3, 4, 5, 6, 7, 8

~9 (8,9,9)

1=~1 (21,11,11)

1=~x ((x+ 1)1,x1,(x 1)1) forx = 2, 3, 4, 5, 6, 7, 8

1=~9 (91,91,81)



7/11

Xni1

Xmj1

Mjgi 1; 1;1 1; 1:15;2:16 1; 1; 1

19:36;25:17;37:33Xni1

Xmj1

Mjgi

" #1

1

37:33;

1

25:17;

1

19:36

0:0267; 0:0397; 0:0516Xmj1

Mjg1 1; 1; 1 1; 1:15; 2:16 1; 1:74; 2:76

5; 6:53; 10:62Xmj1

Mjg2 4:46;5:39; 8:55

Xmj1

Mjg3 3:43;4:75; 6;Xmj1

Mjg4 3:93;4:9; 7:16;

Xmj1

Mjg5 2:54;3:6; 5

F1 Xmj1

Mjg1 Xni1

Xmj1

Mjgi" #1

5; 6:53; 10:62

0:0267; 0:0397;0:0516 0:14;0:31; 0:56

F2 4:46; 5:39; 8:55 0:0267; 0:0397;0:0516

0:10; 0:26;0:48

F3 3:43; 4:75; 6 0:0267; 0:0397; 0:0516

0:05; 0:10;0:23

F4 3:93; 4:9; 7:16 0:0267;0:0397;0:0516

0:07; 0:18;0:34

F5 2:54; 3:6; 5 0:0267; 0:0397;0:0516 0:05; 0:09;0:16

VF1 P F2 1; VF1 P F3 1;

VF1 P F4 1; VF1 P F5 1

VF2 P F1 0:8716; VF2 P F3 1;

VF2 P F4 1; VF2 P F5 1

VF3 P F1 0:7135; VF3 P F2 0:8824;

VF3 P F4 0:9715;

VF3 P F5 1; VF4 P F1 0:7847;

VF4 P F2 0:9280; VF4 P F3 1

VF4 P F5 1; VF5 P F1 0:5170; VF5 P F2 0:6634;

VF5 P F3 0:7850; VF5 P F4 0:7479

The weight vectors are calculated as follows.

dF1 Min VF1 P F2; F3; F4; F5

Min1; 1; 1;1 1dF2 Min VF2 P F1; F3; F4; F5 Min0:8716; 1;1; 1

0:8716

dF3 Min VF3 P F1; F2; F4; F5 Min0:7135; 0:8824;0:9715;1

0:7135

dF4 Min VF4 P F1; F2; F3; F5 Min0:7847; 0:9280;1; 1

0:7847

dF5 Min VF5 P F1; F2; F3; F4

Min0:5170;0:6634; 0:7850; 0:7479 0:5170

W0 dF1;dF2; dF3; dF4;dF5T

1; 0:8716;0:7135;0:7847;0:5170T

0:257; 0:224; 0:184;0:202;0:133

From the above fuzzy-AHP analysis, it is observed that cost has the

highest weight for supplier selection. The weights of quality, GHG

emission, lead time and demand come after that. The management

of ABC Company expressed that the quality and lead time are

important factors for supplier selection. They also commented that

green house gas emission per product is given importance for sup-

plier selection because these would help the company to minimize

their carbon footprint.

5.1. Fuzzy linear programming

In this supplier selection model, we considered four suppliers.

The purchasing criteria such as cost, quality rejection, late delivery

and green house gas emission per product are considered in this

model. Capacity constraint, budget constraint and total purchasing

carbon cap (Ccap) are considered as constraints in this model. These

constraints are deterministic in nature. We have considered de-

mand as a fuzzy variable. The demand is predicted to be about

20,000, and it is assumed that it can vary from 19,950 to 20100.

TheCcap value is taken 30,000 in this model. Supplier quantitative

information is given inTable 3.

Numerical example of multi objective linear programming is gi-

ven below. ObjectiveZ1minimizes the total purchasing cost of the

material. ObjectiveZ2minimizes the rejection due to quality prob-

lem of the product. Objective Z3 minimizes the number of late

delivered item. Objective Z4 minimizes the total carbon footprint

of the purchased item.

Z1 6x1 7x2 4x3 3x4

Z2 0:05x1 0:03x2 0:02x3 0:04x4

Z3 0:03x1 0:02x2 0:08x3 0:04x4

Z4 1:3x1 1:5x2 1:2x3 1:6x4

Subject to;

x1x2x3x4 20; 000

x1 6 6000

x2 6 14;500

x3 6 7000x4 6 4000

1:3x1 1:5x2 1:2x3 1:6x4 6 30; 000

6x1 6 24; 000

7x2 6 70;000

4x3 6 60;000

3x4 6 10;000

x1 P 0; x2 P 0; x3 P 0; x4 P 0

x1;x2;x3;x4 are integer

According to computational procedure discussed earlier, the objec-

tive functionZ1is minimized using the set of constraints for getting

the lower-bound of the objective function. The same objective func-

tion (Z1) is again maximized using the same set of constraints forgetting the upper-bound of the objective function. This procedure

Table 2

Fuzzy pair wise comparisons among the criteria.

Cost Quality Lead time GHGE Demand

Cost (1,1,1) (1,1.15,2.16) (1,1.32,2.35) (1,1.32,2.35) (1,1.74,2.76)

Quality (0.46, 0.87, 1) (1, 1, 1) (1, 1 2) (1, 1, 2) (1, 1.52,

2.55)

Lead

time

(0.43, 0.75, 1) (0.5, 1, 1) (1, 1, 1) (0.5, 1, 1) (1, 1, 2)

GHGE (0.43, 0.75, 1) (0.5, 1, 1) (1, 1, 2) (1, 1, 1) (1, 1.15, 2.16)Demand (0.36,0.57,1) (0.39,0.66,1) (0.33,0.5,1) (0.46,0.87,1) (1,1,1)



8/11

is repeated for rest three objective functions (Z2,Z3 andZ4) for get-

ting the lower and upper bound of these objective functions. The

minimum and maximum value of cost, quality rejection, late deliv-

ery and GHGE are presented inTable 4.

The crisp formulation of the supplier selection problem is for-mulated using the weighted additive model proposed by Tiwari

et al. (1987)(49)(55). The weights calculated by fuzzy-AHP are

used for crisp formulation of the supplier selection problem. In

the crisp formulation, the additive value of membership functions

of the objectives and the constraints is maximized. In crisp formu-

lation, first four terms are the membership functions of the objec-

tive functions (Z1,Z2,Z3 and Z4) and the fifth term (c1) is themembership function of the demand constraint.

Approach 1: hybrid approach

Crisp formulation for supplier selection problem

Maximise 0:257 k1 0:224 k2 0:184 k3

0:202 k4 0:133 c1Subject to;

k1 6118; 000 6x1 7x2 4x3 3x4

16333:3

k2 6686 0:05x1 0:03x2 0:02x3 0:04x4

126:6667

k3 6926 0:03x1 0:02x2 0:08x3 0:04x4

260

k4 628;733 1:3x1 1:5x2 1:2x3 1:6x4

1633:33

c1 620; 100 x1x2x3x4

100

c1 6x1x2x3x4 19; 950

50

x1 6 6000

x2 6 14; 500

x3 6 7000

x4 6 4000

1:3x1 1:5x2 1:2x3 1:6x4 6 30;000

6x1 6 24; 000

7x2 6 70; 000

4x3 6 60; 000

3x4 6 10; 000

x1 P 0; x2 P 0; x3 P 0; x4 P 0 and

x1;x2;x3 and x4 integer:

Linear programming based software LINGO (Ver. 11) has been usedto solve this problem. The optimal solution for the above formula-

tion is obtained as follows.

Objective value is k= 0.6336, and the value of k1= 0.755,

k2= 0.9684,k3= 0.1513, k4= 0.3059, k5= 1, and the value ofx1= 0,

x2= 9667, x3= 7000, x4= 3333.

Z1 105; 668; Z2 563; Z3 887; Z4 28; 233

Supplier selection problem is again solved by Zimmermann (1978)

approach. In Zimmermann approach the weights of the all member-

ship functions is considered same. In this approach k is considered

the overall membership function for all the objective functions

(Z1,Z2,Z3andZ4) and the constraints. The overall membership func-

tion (k) is maximized in this case.

Approach 2: Zimmermann approach

Maximise k

Subject to;

k 6118; 000 6x1 7x2 4x3 3x4

16333:3

k 6686 0:05x1 0:03x2 0:02x3 0:04x4

126:6667

k 6926 0:03x1 0:02x2 0:08x3 0:04x4

260

k 628; 733 1:3x1 1:5x2 1:2x3 1:6x4

1633:33

k 620; 100 x1x2x3x4

100

k 6

x1x2x3x4 19; 950

50

x1 6 6000

x2 6 14; 500

x3 6 7000

x4 6 4000

1:3x1 1:5x2 1:2x3 1:6x4 6 30;000

6x1 6 24; 000

7x2 6 70; 000

4x3 6 60; 000

3x4 6 10; 000

x1 P 0; x2 P 0; x3 P 0; x4 P 0;

and x1;x2;x3;x4 integer

We get k = 0.4586 andx1= 2922,x2= 9088, x3= 5469,x4= 2494.

Z1 110; 506; Z2 628; Z3 806; Z4 27; 981

The solutions are summarized in Table 5. It is observed that for a

range of demand between 19,950 and 20,100, the optimized cost,

net rejection, late delivered item and GHGE is $105,668, 563 unit,

887 unit, and 28,233 kg respectively. When this problem is solved

using Zimmermann approach, the optimized cost, net rejection, late

delivered item and GHGE is $110,506, 628 unit, 806 unit and

27,983 kg respectively. In our proposed approach, the quota alloca-

tion to Supplier 1 is zero. However, in case of Zimmermann ap-

proach 14.6 percent quota is allocated to Supplier 1. Table 6

shows that the quota allocation to Suppliers calculated by two dif-ferent methods. It is observed that thequota allocation to Supplier 1

Table 5

Comparison between Hybrid and Zimmermann approach.

SN Objective function Hybrid approach Zimmermann approach

1 Z1 105,668 110,506

2 Z2 563 628

3 Z3 887 806

4 Z4 28,233 27,983

Table 3

Suppliers quantitative information.

Supplier Pi($) Qi(Percentage)

Li(Percentage)

GHGE

(kg)

Ui Bi($)

1 6 0.05 0.03 1.3 6000 24,000

2 7 0.03 0.02 1.5 14,500 70,000

3 4 0.02 0.08 1.2 7000 60,000

4 3 0.04 0.04 1.6 4000 10,000

Table 4

The data set for membership function calculation.

Serial number Objective function l= 1 l= 0

1 Z1 101666.7 118,000

2 Z2 560 686.6667

3 Z3 666.6667 926.6667

4 Z4 27,100 28733.33



9/11

varies significantly for two methods. Supplier 1 has lost its entire

quota because the product cost of Supplier 1 is higher than the

other suppliers. The quality supplied by Supplier 1 is much more

inferior to other suppliers. In our proposed approach, more stress

is given on cost, quality and carbon footprint for selecting the sup-plier but in Zimmermann approach all variables are given same

weights. Thats why we observed the different quota allocation to

the suppliers for these two approaches. 35 percent quota is allo-

cated to Supplier 3 when the calculation is done using hybrid ap-

proach, and 27.4 percent quota is allocated to supplier 3 when it

is solved by Zimmermann approach. Supplier 3 has received the

maximum order as compared to the supplying capacity of the sup-

pliers. The maximum order is due to its lower cost product, lowest

quality rejection and lower GHGE. Late delivery is given lower

weight in this model that enhances the order quantity to Supplier

3. Supplier 4 is allocated 16.4 percent quota in spite of the lowest

cost of product among the suppliers. The lower quota allocation is

due to highest carbon footprint of supplied products. The remaining

quota is fulfilled by Supplier 2. It is observed that 48.3 percent quo-ta has been allocated to supplier 2. It has the highest quota alloca-

tion than other suppliers. The highest quota allocation is due to the

lower quality rejection percentage and highest supplying capacity.

Supplier 3 is ranked the best on the basis of low cost, lowest quality

rejection and lower GHGE. However, the supplier does not have thesufficient capacity to supply.

Table 6

Suppliers quota allocation.

Supplier Ui Solution using

hybrid approach

Quota allocation

percentage for Hybrid

Solution using

Zimmermann approach

Quota allocation percentage

for Zimmermann

1 6000 0 0 2922 14.6

2 14,500 9667 48.3 9088 45.5

3 7000 7000 35 5469 27.4

4 4000 3333 16.7 2494 12.5

Fig. 3. Variation of achievement of cost goal (k1) and quality goal (k2) with respect to overall goal (k).

Fig. 4. Variation of achievement of lead time goal (k3) and GHGE goal (k4) with respect to overall goal (k).

Fig. 5. Variation of achievement of demand fulfillment (c1) goal with respect tooverall goal (k).



10/11

Variability of the individual goal achievement of hybrid model

is checked by fixing the value of objective function. Fig. 3 shows

the individual variability of achievement of cost goal (k1) and qual-

ity goal (k2) with respect to total achieved goal (k).Fig. 4 shows the

fluctuation of lead time goal (k3) and GHGE goal (k4) with respect

to the overall achieved goal (k). Fig. 5 shows the variability of

demand fulfillment goal (c1) with respect to the overall achievedgoal (k). Fig. 6 shows the quota allocations to the suppliers. It is

observed that quota allocation to different suppliers is different

for different (k) values.

As the degree of the achievement of these fuzzy goals changes,

the quota allocation to the suppliers also changes. It is observed

that quota allocation to Supplier 1 is not changing, when k value

is changing from 0 to 0.52, after that it drops to 0 and then again,

comes back to the value 4000. For k value ranging from 0.54 to

0.62, there is no change in supplier quota allocation to Supplier

1. After k value of 0.62 the quota allocation to Supplier 1 is de-creased to 0. Quota allocation to Supplier 4 is zero up to k value

0.46, and then the allocated quota is increased and stabilized up

tok value 0.6336. For k value 0.56 the quota allocation to Supplier

4 is zero. Supplier 2 and Supplier 3 follow the mixed trend, and

their quota allocations are changing according to the value of k.

Fork value 0.6336 the quota allocation to Supplier 2 is 9667 and

for the same k value quota allocation to Supplier 3 is 7000.

In our proposed approach, the degree of achievement of late

delivered item goal (k3) is obtained as 0.1513. This achievement le-

vel may not be sufficient to satisfy decision makers objective of

lesser late delivered items. In the realistic situation, one cannot ar-

gue that a poor performance of one criterion can be balanced with

a good performance of other criteria. To solve this problem, the

presented model is again reformulated by including an additionalcondition such that the achievement level of membership func-

tions should not be less than an allowed value (Amid et al.,

2006). The a-cut approach is used for reformulation of the prob-lem. The a-cut approach ensures that the degree of achievementof any goal is more than the specified value ofa. The weightedadditive model is to be reformulated by incorporating new

constraints,kjP aandckP aand a 2 [a,a+] to other system con-

straints (Amid et al., 2006). In this approach, an appropriate value

ofa is to be chosen to avoid the infeasible solutions (Chen, 1985).After discussing this issue with decision makers, we have reformu-

lated the model considering the following constraintsk1> 0.5;k2> 0.5;k3> 0.4;k4 > 0.4 and c1= 1. After solving the equation, weobtained the maximum achievable goal which is k= 0.5432. The

individual achievable goals are k1= 0.5002; k2= 0. 5684;k3= 0.4;k4= 0.4 and c1 = 1. The final solution is obtained as: x1= 2264;

x2= 9316; x3= 5774 and x4= 2646. The optimized cost is

$109,830, rejection due to quality problem is 614, late delivered

item is 822 and carbon emission is 28,079 kg.

6. Conclusions

By nature, supplier selection process is a complicated task. The

profitability and customer satisfaction are directly proportional to

the effectiveness of supplier selection. Therefore, supplier selection

is a crucial strategic decision for long term survival of the firm. In

the present supplier selection model, a combined approach of fuz-

zy-AHP and fuzzy multi-objective linear programming are used.

This formulation integrates carbon emission in the objective func-

tion, and carbon emission cap (Ccap) of sourcing as a constraint

while selecting a supplier. In this model fuzzy AHP is used first

to calculate the weights of the criteria and then fuzzy linear pro-gramming is used to find out the optimum solution of the problem.

Vagueness and imprecision can be effectively handled in this mod-

el. The proposed model is very useful to solve the practical prob-

lem. In the practical situation, all objective functions do not

possess same weights therefore; the weights of the objective func-

tions can be changed according to the requirement of the manager.

The individual priority can be easily calculated by using the fuzzy-

AHP method. The proposed method is a very useful decision-mak-

ing tool for mitigating environmental challenges. A case study has

been used to demonstrate the implication of fuzzy-AHP and fuzzy

linear programming for supplier selection problem.

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